Compound Bursting Behaviors in the Parametrically Amplified Mathieu–Duffing Nonlinear System

Ma, Xindong; Zhang, Xiaofang; Yu, Yue; Bi, Qinsheng

doi:10.1007/s42417-021-00366-y

Cited by 7 publications

(2 citation statements)

References 57 publications

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“…Bursting oscillations, produced by the coupling of distinct timescale effects, behaving in the switching of different oscillations with amplitudes, are common found in many classical nonlinear systems, such as preypredator model [1,2], circuit oscillators [3,4], energy harvesters [5,6] and neurodynamic systems [7,8]. The switching of the system between oscillations of different amplitudes is essentially a transmission process of information exchange, which has been confirmed from the neuronal experiments [9,10].…”

Section: Introductionmentioning

confidence: 70%

On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

Zhang,

Tang

2023

Phys. Scr.

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Multiple timescale effects can be reflected bursting oscillations in many classical nonlinear oscillators. In this work, we are concerned about the bursting oscillations induced by two timescale effects in the damped Helmholtz-Rayleigh-Duffing oscillator (written as DHRDO for short) excited by slow-changing parametrical and external forcings. By using trigonometric function variation and authenticating the slow excitations as a slowly varying state variable, the time-varying DHRDO can be rewritten as a new time-invariant system. Then, the critical conditions of some typical bifurcations are presented by bifurcation theory. With the help of bifurcation analyses, six bursting patterns, i.e., ‘Hopf/Hopf-Hopf/Hopf’ bursting, ‘fold/Homoclinic-Hopf/Hopf’ bursting, ‘fold/Homoclinic/Hopf’ bursting, ‘Hopf/fold/Homoclinic/Hopf’ bursting, ‘Hopf/Homoclinic/Homoclinic/Hopf’ bursting and ‘Hopf/Homoclinic/Hopf-Hopf/Homoclinic/Hopf’ bursting, are explored by the slow/fast decomposition method and the other techniques. Our findings provide different forms of the excited state oscillation modes as well as the bursting patterns. In addition, we use the numerical simulation to prove the correctness of the theoretical analyses.

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Section: Introductionmentioning

confidence: 70%

On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

Zhang,

Tang

2023

Phys. Scr.

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“…Bursting behaviors, as a typical fast-slow dynamical behavior, displaying in a special vibration mode that is presented as the coupling of the repetitive-spiking oscillations and silent-state oscillations, are often observed in many nonlinear models, such as Hindmarsh-Rose model [1,2], Rayleigh-van der Pol-Duffing oscillator [3,4], Duffing-van der Pol system [5,6] and Mathieu-Duffing equation [7,8]. Normally, the repetitive-spiking oscillations correspond to the trajectories moving in the big-amplitude limit cycles and the silent-state oscillations agree with the trajectories vibrating in the equilibrium domain or the small-amplitude limit cycles.…”

Section: Introductionmentioning

confidence: 99%

Analysis on the symmetric fast-slow behaviors in a van der Pol-Duffing-Jerk oscillator

Lyu,

Li,

Huang

et al. 2023

Phys. Scr.

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The study of bursting oscillations induced by frequency-domain multiscale effect is one of the key scientific issues in the theoretical analysis of circuit systems. In this paper, we explore the mechanism of the bursting oscillations of a van der Pol-Duffing-Jerk circuit oscillator with slow-changing parametric and external periodic excitations. Three typical bursting modes, namely, left-right symmetric “subHopf/fold limit cycle” bursting, origin symmetric “fold/fold limit cycle” bursting and origin symmetric “fold/subHopf/fold limit cycle” bursting, are presented. The slowly changing excitation is treated as a generalized state variable to analyze the influence on the critical manifolds of the equilibria and bifurcations. The critical conditions of fold and Hopf bifurcations are computed by using the bifurcation theory, and two typical bifurcation structures are obtained according to the position of different bifurcation curves. Based on the bifurcation analysis, we investigate the appearance and dynamicalal evolutions of the different bursting oscillations with the variation of the external excitation amplitude. It is pointed that not only the bifurcation structures but also the distance between the fold and Hopf bifurcation points can affect the bursting patterns. We find the directions of the trajectories and the bursting types are sensitive to the values of the external excitation amplitude. Furthermore, we reveal the mechanism of the bursting oscillations by overlapping the trajectories on (,x)-plane onto the corresponding bifurcation structures. Numerical simulations are also presented to prove the correctness of the theoretical analysis in our study.

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Occurrence of mixed-mode oscillations in a system consisting of a Van der Pol system and a Duffing oscillator with two potential wells

Lyu,

Li,

Huang

et al. 2024

Nonlinear Dyn

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Compound Bursting Behaviors in the Parametrically Amplified Mathieu–Duffing Nonlinear System

Cited by 7 publications

References 57 publications

On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

On the occurrence of bursting oscillations in the damping Helmholtz–Rayleigh–Duffing oscillator with slow-changing parametrical and external forcings

Analysis on the symmetric fast-slow behaviors in a van der Pol-Duffing-Jerk oscillator

Occurrence of mixed-mode oscillations in a system consisting of a Van der Pol system and a Duffing oscillator with two potential wells

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